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Maxwell’s Ekvationer
Fö10
2016
⋅dA =
Gauss Lag för Elektriska Fältet
∫E
Gauss Lag för Magnetiska Fältet
∫B ⋅dA
Faraday’s Lag
∫E
qenc
ε0
=0
⋅ds = −
dΦ E
+ µ 0 ienc
dt
Ampere – Maxwell’s Lag
∫B ⋅ds
Lorentz Kraften
F =q ⋅E + qv×B
= µ0 ε 0
dΦ B
dt
Maxwell’s Ekvationer
Elektronens Magnetiska Dipolmoment
Physics Handbook
∫E
⋅dA =
qenc
∫B ⋅dA =0
∫E
⋅ds = −
∫ B ⋅ d s = µ0 ε 0
div E = ∇ ⋅ E =
ε0
dΦ B
dt
dΦ E
+ µ0 ienc
dt
qenc
ε0
Orbital Magnetic
div B = ∇ ⋅ B = 0
curl E = ∇ × E = −
Dipol Moment
dB
dt
dE
curl B = ∇ × B = µ0 j + µ 0 ε 0
dt
Spin Magnetic
µ orb = −
µS = −
Dipol Moment
e
L orb
2m
e
S
m
Diamagnetism
Svaga magnetiska dipoler skapas av
externt magnetfält
Paramagnetism
Permanenta magnetiska dipoler ställer in
sig i extern magnetfält
Ferromagnetism Starka permanenta magnetiska dipoler
påverkar varandra. Permanent magneter
32.9: Diamagnetism:
1. Diamagnetism: In diamagnetism, weak magnetic dipole moments are produced in the
atoms of the material when the material is placed in an external magnetic
field Bext; the combination gives the material as a whole only a feeble net
magnetic field.
If a magnetic field is applied, the
diamagnetic material develops a
magnetic dipole moment and experiences
a magnetic force. When the field is
removed, both the dipole moment and the
force disappear.
32.10: Paramagnetism:
Each atom of such a material has a permanent resultant magnetic dipole moment, but the
moments are randomly oriented in the material and the material lacks a net magnetic field.
An external magnetic field Bext can partially align the atomic magnetic dipole moments to
give the material a net magnetic field.
The ratio of its magnetic dipole moment to its volume V. is
the magnetization M of the sample, and its magnitude is
In 1895 Pierre Curie discovered experimentally that the
magnetization of a paramagnetic sample is directly
proportional to the magnitude of the external magnetic field
and inversely proportional to the temperature T.
is known as Curie’s law, and C is called the Curie constant.
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32.11: Ferromagnetism:
32.11: Ferromagnetism: Hyteresis
Some of the electrons in these materials have their resultant magnetic dipole moments
aligned, which produces regions with strong magnetic dipole moments. An external field Bext
can align the magnetic moments of such regions, producing a strong magnetic field for the
material.
Magnetization curves for ferromagnetic materials are
not retraced as we increase and then decrease the
external magnetic field B0.
Figure 32-18 is a plot of BM versus B0 during the
following operations with a Rowland ring:
If the temperature of a ferromagnetic material is raised above a certain critical
value, called the Curie temperature, the exchange coupling ceases to be effective.
Most such materials then become simply paramagnetic.
I.Starting with the iron unmagnetized (point a),
increase the current in the toroid until B0 (=µ0in) has
the value corresponding to point b;
II.reduce the current in the toroid winding (and thus
B0) back to zero (point c);
III.reverse the toroid current and increase it in
magnitude until B0 has the value corresponding to
point d;
IV.reduce the current to zero again (point e);
V.reverse the current once more until point b is
reached again.
The lack of retraceability shown in Fig. 32-18 is called hysteresis, and the curve bcdeb is
called a hysteresis loop.
32.6: Magnets: The Magnetism of Earth:
Ch. 33 Electromagnetic Waves
• EM vågor matematiskt m.h.a. Maxwells ekvationer
• Polarisation
• Refraktion och Reflektion
• Chromatisk Dispersion
• Total Internal Reflection
(33-1)
Maxwell’s Rainbow
Mathematical Description of Traveling EM Waves
The wavelength/frequency range in which electromagnetic (EM) waves (light) are
visible is only a tiny fraction of the entire electromagnetic spectrum.
Fig. 33-5
Electric Field: E
= E m sin ( kx − ω t )
Magnetic Field:B
= Bm sin ( kx − ω t )
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Mathematical Description of Traveling EM Waves
Electric Field: E
k=
Utbredningshastighet
Vacuum Permittivity:
Vacuum Permeability:
v=
ω
k
=
Changing magnetic fields produce electric fields, Faraday’s law of induction:
2π
λ
Angular frequency, vinkelfrekvens:
Amplitude Ratio:
Induced Electric Field
= Em sin ( kx − ω t )
Wavenumber, Vågtal :
λ
τ
ω=
=
dΦ B
dt
∫E
⋅ds = −
2π
∫E
⋅ d s = ( E + dE ) h − Eh = h dE
τ
Φ B = B A = B hdx
c=
1
µ0ε 0
ε0
⇒ h dE = − h dx
⇒
µ0
Em
=c
Bm
The Traveling EM Wave, Quantitatively
Magnitude Ratio:
E (t )
=c
B (t )
The Traveling EM Wave, Quantitatively
dB
dt
⇒
dE
dB
=−
dx
dt
∂E
∂B
=−
∂x
∂t
∂E
∂B
= kEm cos ( kx − ω t ) and
= −ω Bm cos ( kx − ω t )
∂x
∂t
kEm cos( kx − ωt ) = + ωBm cos( kx − ωt ) ⇒
Em ω
= =c
Bm k
Polarization
Induced Magnetic Field
Changing electric fields produce magnetic fields, Maxwell’s law of induction:
∫ B ⋅ ds
= µ0 ε 0
dΦ E
dt
∫ B ⋅ d s = − ( B + dB) h + Bh = − h dB
dΦ E
dE
= h dx
dt
dt
dB
⇒ − h dB = µ 0 ε 0 ( h dx
)
dt
∂B
∂E
⇒−
= µ0 ε 0
∂x
∂t
The polarization of light describes
how the electric field in the EM
wave oscillates.
Φ E = E A = E hdx ⇒
Fig. 33-7
Vertically planepolarized (or linearly
polarized)
− kBm cos ( kx − ω t ) = − µ0ε 0ω Em cos ( kx − ω t )
Em
1
1
=
=
=c→c=
Bm µ0ε 0 (ω k ) µ0ε 0c
1
(33-13)
µ0ε 0
Polarized Light
Unpolarized or randomly polarized light
has its instantaneous polarization direction
vary randomly with time.
Reflection and Refraction
Although light waves spread as they move from a source, often we can
approximate its travel as being a straight line → geometrical optics.
What happens when a narrow beam of
light encounters a glass surface?
Law of Reflection
One
Reflection: θ1 ' = θ1
Only the electric field component
along the polarizing direction of
polarizing sheet is passed
(transmitted); the perpendicular
component is blocked (absorbed).
Snell’s Law
Refraction:n2 sin θ 2
sin θ 2 =
= n1 sin θ1
n1
sin θ1
n2
n is the index of refraction of the material. Brytningsindex
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Waves
For light going from n1 to n2:
•
n2 = n1 → θ2 = θ1
•
n2 > n1 → θ2<θ1, light bent toward normal
•
n2 < n1 → θ2 >θ1, light bent away from normal
Chromatic Dispersion
The index of refraction n encountered by light in
any medium except vacuum depends on the
wavelength of the light. So if light consisting of
different wavelengths enters a material, the
different wavelengths will be refracted
differently → chromatic dispersion.
Total Internal Reflection
Rainbows
For light that travels from a medium with a larger index of refraction to a medium
with a smaller index of refraction n1 > n1 → θ2 > θ1, as θ1 increases, θ2 will reach
90o (the largest possible angle for refraction) before q1 does.
n2
n1 sin θ c = n2 sin 90° = n2
Critical Angle:θ c
n1
Fig. 33-22
= sin −1
n2
n1
When θ2 > θc no light is
refracted (Snell’s law does not
have a solution!) so no light is
transmitted → Total Internal
Reflection
Light as a Wave
Ch. 35 Interference
Huygen’s Principle: All points on a wavefront serve as point sources of spherical
secondary wavelets. After time t, the new position of the wavefront will be that of a
surface tangent to these secondary wavelets.
• Hyugens principle
• Interferens
• Youngs Dubbelspalt
• Diffraktion
• Interferens i tunna filmer
(35-2)
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Law of Refraction
t=
sin θ1 =
sin θ 2 =
λ1
λ1
v1
=
λ2
v2
→
(for triangle hce)
hc
λ2
hc
P
När två eller flera vågor samverkar
uppstår interferens.
Två vågor med samma w samverkar i
punkten P
sin θ1 λ1 v1
=
=
sin θ 2 λ2 v2
(for triangle hcg )
Index of Refraction:
c
n1 =
v1
Interferens
λ1 v1
=
λ2 v2
n=
S2
Faskillnaden mellan dessa blir δ = kr1 - kr2 = (r1 - r2 )*2π/λ
Totala amplituden i P blir (10.3)
c
and n2 =
v2
Law of Refraction:
r2
S1
Y1 = A1 sin ( kr1 – wt )
Y2 = A2 sin ( kr2 – wt )
c
v
Amax då cos δ = 1,
sin θ1 c n1 n2
=
=
sin θ 2 c n2 n1
Fig. 35-3
r1
eller
Amin då cos δ = -1,
n1 sin θ1 = n2 sin θ 2
eller
Interference in different media
A = A12 + A22 + 2 A1 A2 cos δ
dvs δ = n 2π
(r1 - r2 ) = n λ
Konstruktiv Interferens
dvs δ = (2n+1)π
(r1 - r2 ) = (2n+1)λ/2
Destruktiv Interferens
Diffraction
v
cn c
fn =
=
= = f
λn λ n λ
Frekvensen hos ljuset i ett
medium är samma som i vacuum.
Eftersom utbredningshastigheten
ändras måste våglängden ändras.
λn v
v
λ
= → λn = λ → λn =
c
n
λ c
For plane waves entering a single slit, the waves emerging from the slit
start spreading out, diffracting.
Since wavelengths in n1 and n2 are different, the
two beams may no longer be in phase.
Fig. 35-4
Number of wavelengths in n1: N 1 =
N um ber of w aveleng th s in n 2 : N 2 =
Assuming n2 > n1: N 2 − N1 =
Ln2
λ
−
Ln2
λ
=
L
λ
L
λ n1
=
L
λn 2
L
λ n1
=
=
Ln1
λ
L
L n2
=
λ n2
λ
( n2 − n1 )
N 2 − N1 = 1/2 wavelength → destructive interference
(35-4)
Young’s Experiment
For waves entering two slits, the emerging waves interfere and form
an interference (diffraction) pattern.
Young’s Experiment
For waves entering two slits, the emerging waves interfere and form
an interference (diffraction) pattern.
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Interference in Double Slit
Interference in Double Slit
The phase difference between two waves can change if the waves travel paths of
different lengths.
What appears at each point on the screen is determined by the path length
difference DL of the rays reaching that point.
D >> d
Konstruktiv interferens, dvs ljust band, då skillnaden i
gångssträcka är jämna multiplar av våglängden. Fasskillnad jämna
multiplar av π
∆L = d sinΘ = mλ m=0,1,2,3,,,
Destruktiv interferens, mörka band, då skillnaden i gångsträcka är
halva våglängder. Fasskillnad udda multiplar av π
∆L = d sinΘ = (2m+1)λ/2 m=0,1,2,3,,,
Path Length Difference: ∆L
= d sin θ
Reflection Phase Shifts
Interference from Thin Films
n1
φ12 = ?
n1
n1 > n2
n1 < n2
n2
n2
Reflection
Off lower index
Off higher index
Reflection Phase Shift
0
0.5 wavelength
θ ≈0
Fig. 35-16
Då en våg går från medium med lågt
brytningsindex (lättare medium) till ett
medium med högre brytningsindex får
man ett fasskift motsvarande
• 0.5 våglängder
• π eller 180º
(35-14)
(35-15)
Reflection Interference in thin Films
Three effects can contribute to the phase
difference between r1 and r2.
1. Differences in reflection conditions.
B
0
λ A
2
3. Differences in the media in which the waves
travel. One must use the wavelength in each
medium (l / n) to calculate the phase.
Normally phaseshift at A, but not at B.
Konstruktiv interferens, ljust band, då skillnaden i gångsträcka
2L = ( m +
1
2
)
λ
n2
Ch. 36 Diffraktion
2. Difference in path length traveled.
•Diffraktion
• Diffraktion i Enkelspalt
• Diffraktionsgitter
• XRAY Diffraktion
for m = 0,1, 2,K (maxima-- bright film in air)
Destruktiv interferens, mörkt band, då skillnaden i gångsträcka
2L = m
λ
n2
for m = 0,1, 2,K (minima-- dark film in air)
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Diffraction and the Wave Theory of Light
Diffraktion i Enkel Spalt: Intensitetsminima
Diffraction pattern from a single narrow slit.
Side or secondary
maxima
Light
Central
maximum
These patterns cannot be
explained using geometrical
optics (Ch. 34)!
Fresnel Bright Spot.
Light
Bright
spot
(36-2)
Diffraktion i Enkel Spalt: Intensitetsminima
Diffraction by a Single Slit: Locating the Minima, cont'd
Gör sedan samma argumentation för interferens
mellan strålgångar från kanten och från a/4,
vilket ger andra minimum.
Jämför vågor som går från kanten
respektive mitten av spalten, dvs avstånd
a/2.
a
λ
sin θ = → a sin θ = 2λ (second minimum)
4
2
P.s.s. som för dubbelspalten antar man då
D>>a att dessa blir parallella och skillnaden
I gångsträcka
L = (a/2) sinΘ
Till slut får vi interferens minimum vid
Destruktiv interferens, intensitetsminima,
då L = λ/2 = (a/2) sinΘ.
a sin θ = mλ , for m = 1, 2, 3K
dvs λ = a sinΘ
(minima-dark fringes)
Alla motsvarande strålgångar genom
spalten ger samma bidrag.
Fig. 36-5
Diffraction Gratings
A device with N slits (rulings) can be used to manipulate light, such as
separate different wavelengths of light that are contained in a single
beam. How does a diffraction grating affect monochromatic light?
Width of Lines
The ability of the diffraction grating to resolve (separate) different
wavelengths depends on the width of the lines (maxima).
Fig. 36-20
Fig. 36-17
Fig. 36-18
d sin θ = mλ for m = 0,1, 2 K (maxima-lines)
Fig. 36-19
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Grating Spectroscope
Separates different wavelengths (colors) of light into
distinct diffraction lines
X-Ray Diffraction
X-rays are electromagnetic radiation with wavelength ~1 Å = 10-10 m
(visible light ~5.5x10-7 m).
X-ray generation
Fig. 36-23
X-ray wavelengths too short to be resolved
by a standard optical grating
Reflektions gitter
Fig. 36-27
θ = sin −1
(1)( 0.1 nm ) = 0.0019°
mλ
= sin −1
d
3000 nm
Fig. 36-22
X-Ray Diffraction
Diffraction of x-rays by crystal: spacing d of
adjacent crystal planes on the order of 0.1 nm
→ three-dimensional diffraction grating with
diffraction maxima along angles where
reflections from different planes interfere
constructively
2d sin θ = mλ for m = 0,1, 2 K (Bragg's law)
Fig. 36-28
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